chapter 08 im 10th ed

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CHAPTER 8

Stock Valuation

CHAPTER ORIENTATION

This chapter continues the introduction of concepts underlying asset valuation began inChapter 7. We are specifically concerned with valuing preferred stock and common stock.We also look at the concept of a stockholder’s expected rate of return on an investment.

CHAPTER OUTLINE

I. Preferred Stock

A. Features of preferred stock

1. Owners of preferred stock receive dividends instead of interest.

2. Most preferred stocks are perpetuities (non-maturing).

3. Multiple classes, each having different characteristics, can be issued.

4. Preferred stock has priority over common stock with regard to claimson assets in the case of bankruptcy.

5. Most preferred stock carries a cumulative feature that requires all pastunpaid preferred stock dividends to be paid before any common stock dividends are declared.

6. Preferred stock may contain other protective provisions, such asgranting voting rights in the event of non-payment of dividends.

7. Preferred stock may contain provisions to convert to a predeterminednumber of shares of common stock.

8. Some preferred stock contains provisions for an adjustable rate of return.

9. If there is a participation feature, it allows preferred stockholders to participate in earnings beyond the payment of the stated dividend.

10. Payment-in-kind (PIK) preferred stock, grants the investor additional preferred stock instead of dividends for a given period of time.Eventually cash dividends are paid.

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11. Retirement features for preferred stock are frequently included.

a. Callable preferred refers to a feature which allows preferredstock to be called, or retired, like a bond.

b. A sinking fund provision requires the firm periodically set asidean amount of money for the retirement of its preferred stock.

B. Valuation of preferred stock (V ps):

The value of a preferred stock equals the present value of all future dividends.If the stock is nonmaturing, where dividends are expected in equal amounteach year in perpetuity, the value may be calculated as follows:

V ps = =

II. Common Stock

A. Features of Common Stock

1. As owners of the corporation, common shareholders have the right to

the residual income and assets after bondholders and preferredstockholders have been paid.

2. Common stockholders are generally the only security holders with theright to elect the board of directors.

3. Preemptive rights (if granted) entitle the common shareholder tomaintain a proportionate share of ownership in the firm.

4. Common stockholder’s liability as an owner of the corporation islimited to the amount invested in the stock.

5. Common stock’s value is equal to the present value of all future cashflows expected to be received by the stockholder.

B. Valuing common stock

1. Company growth occurs by:

a. the infusion of new capital, or

b. the retention of earnings, which is called internal growth. Theinternal growth rate of a firm equals:

Return on equity ×

2. Although the bondholder and preferred stockholder are promised aspecific amount each year, the dividend for common stock is based on

the profitability of the firm and management's decision either to paydividends or retain profits for reinvestment.

3. The common dividend typically increases with the growth in corporateearnings.

4. The earnings growth of a firm should be reflected in a higher price for the firm's stock.

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5. In finding the value of a common stock (Vcs), we should discount all

future expected dividends (Dl, D2, D3, ..., D∞) to the present at the

required rate of return for the stockholder (k cs). That is:

Vcs =1

cs

1

)k (1

D

++

2

cs

2

)k (1

D

++...+

∞∞

+ )k (1

D

cs

6. If we assume that the amount of dividend is increasing by a constantgrowth rate each year,

Dt = D0 (l + g)t

where g = the growth rate

D0 = the most recent dividend payment

If the growth rate, g, is the same each year, t, and is less than therequired rate of return, k cs, the valuation equation for common stock

can be reduced to

Vcs = =

III. Shareholder's Expected Rate of Return

A. The shareholder's expected rate of return is of great interest to financialmanagers because it tells about the investor’s expectations.

B. Preferred stockholder's expected rate of return:

If we know the market price of a preferred stock and the amount of thedividends to be received, the expected rate of return from the investment can be determined as follows:

expected rate of return =or

k ps =0P

D

C. Common stockholder's expected rate of return:

1. The expected rate of return for common stock can be calculated fromthe valuation equations previously discussed.

207



2. Assuming that dividends are increasing at a constant annual growthrate, g, we can show that the expected rate of return for commonstock, k cs is

k cs = +

= + g

Since dividend ÷ price is the "dividend yield," the

Expected rate of return = +

IV. Appendix: The Relationship between Value and Earnings

A. Earnings and Value Relationship: The nongrowth firm

1. Nongrowth firms retain no profits for reinvestment purposes.

a. Investments are made to maintain status quo.

b. Earnings and dividend growth stream is constant from year toyear.

2. Value on nongrowth common stock, Vng:

cs

1

cs

1ng

k

D

k

EPS V ==

a. Value of share changes in direct relationship with changes inearnings per share.

b. Changes in the investor’s required rate of return will changeshare value.

B. Earnings and Value Relationship: The growth firm

1. Growth firm reinvests profits back into the business.

2. Value of stock equals the present value of the dividend stream plus the present value of the future growth resulting from reinvesting futureearnings.

NVDG k

EPS V

cs

1cs +=

a. NVDG is the net value of any dividend growth resulting fromreinvestment of future earnings.

b. Present value (PV1) from reinvesting part of the firms earningsin year 1 equals:

)EPS (r k

ROEEPSr PV 1

cs

11 ×−

××=

a. Using the constant-growth model to value NVDG,

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g k

PV NVDG

cs

1

−=

b. The value of a share of stock is therefore:

g k

PV

k

EPS V

cs

1

cs

1cs −

+=

3. Value of stock is influenced by

a. Size of the firm’s EPS,

b. Percentage of profits retained,

c. Spread between return generated on new investments and theinvestor’s required rate of return.

ANSWERS TO

END-OF-CHAPTER QUESTIONS8-1. Preferred stock is often referred to as a hybrid security. This is because preferred

stock has many characteristics of both common stock and bonds. It hascharacteristics of common stock, such as no fixed maturity date, nonpayment of dividends does not force bankruptcy, and the nondeductibility of dividends for tax purposes. But it is like bonds because the dividends are fixed in amount like interest payments. From the point of view of the preferred stockholder, this is not the mostadvantageous combination. On one hand, the dividends are limited as with bondinterest, but the security of forced payment by the threat of bankruptcy is not there.Thus, from the point of view of the investor, the worst features of common stock and bonds are combined.

8-2. To a certain extent, preferred stock dividends can be thought of as a liability. Themajor difference between preferred dividends in arrears and normal liabilities is thatnonpayment of them cannot force the firm into bankruptcy. However, since the goalof the firm is common shareholder wealth maximization, which involves gettingmoney to the common shareholders (dividends), preferred arrearages provide a barrier to achieving this goal.

8-3. A cumulative feature requires all past unpaid preferred stock dividends be paid beforeany common stock dividends are declared. A stockholder would like preferred stock to have a cumulative dividend feature because without it there would be no reasonwhy preferred stock dividends would not be omitted or passed when common stock

dividends were passed. Since preferred stock does not have the dividend enforcement power of interest from bonds, the cumulative feature is necessary to protect the rightsof preferred stockholders.

Other frequent protective features serve to allow for voting rights in the event of nonpayment of dividends or to restrict the payment of common stock dividends if sinking-fund payments are not met or if the firm is in financial difficulty. In effect, the protective features included with preferred stock are similar to the restrictive provisions included with long-term debt.

209



8-4. Fixed rate preferred stock has dividends that do not vary from the fixed amount or from period to period.

Adjustable rate preferred stock is preferred stock that has quarterly dividends thatfluctuate with interest rates under a formula that ties the dividend payment at either a premium or discount to the highest of the three-month Treasury bill rate, the 10-year Treasury bond constant maturity rate, or the 20-year Treasury bond constant maturity

rate. The rates have maximum and minimum levels called the dividend rate band.The purpose of allowing the dividend rate to fluctuate is to minimize the fluctuation inthe value of the preferred stock. It is also very appealing in times of high andfluctuating interest rates.

8-5. With PIK (payment-in-kind) preferred stock, investors receive no dividends initially;they merely get more preferred stock, which in turn pays dividends in even more preferred stock. Usually after 5 or 6 years, if all goes well for the issuing company,cash dividends should replace the preferred stock dividends, generally ranging from12 percent to 18 percent, to entice investors to purchase PIK preferred.

8-6. Convertibility allows a preferred stockholder to convert or exchange preferred stock

for shares of common stock at a predetermined exchange rate. This option gives preferred stockholders more freedom in investment decisions by allowing them toconvert into common stock at their discretion. It gives the preferred stockholder ahigher cash return than the common stock but allows for sharing in some of the futureappreciation of the common stock if they convert the stock.

Preferred stock may be callable by the issuer so that in the event interest rates declineand cheaper funding becomes available, the stock may be called and new securitiesmay be issued at a lower cost. To agree to the call feature, the investor requires aslightly higher rate of return. Call of a convertible preferred stock enables a companyto turn the preferred stock into common equity; i.e., calling it without having to spendthe cash.

8-7. Both values are based on future cash flows to be received by stockholders. Preferredstock typically has a predetermined constant dividend. For common stock, thedividend is based on the profitability of the firm and on management’s decision to paydividends or to retain the profits for reinvestment purposes. Thus, the growth of future dividends is a prime distinguishing feature of common stock.

8-8. The expected rate of return is the rate of return that may be expected from purchasinga security at the prevailing market price. Thus, the expected rate of return is the ratethat equates the present value of future cash flows with the actual selling price of thesecurity in the market.

210



8-9. The required rate of return is the discount rate that equates the present value of futurecash flows with the intrinsic value of the security. As with the internal rate of returnfor a capital budgeting problem, we have to find the rate of return that sets the futurecash flows equal to the cost of the security. This rate may have to be developed bytrial and error.

8-10. The two types of return are dividend income and capital gains. The dividend incomefor common stockholders differs from preferred stockholders, in that no specifieddividend amount is to be received. However, the common stockholders are permittedto participate in the growth of the company. As a result of this growth, their secondsource of return, price appreciation, is realized.

SOLUTIONS TO

END-OF-CHAPTER PROBLEMS

Solutions to Problem Set A

8-1A. Value (V ps) =12.

6$

= $50.00

8-2A. Growth rate = return on equity x retention rate

= (16%) × (60%) = 9.6%

8-3A. Value (V ps) =12.

100$ 14. ×

= 12.

14$

= $116.67

8-4A. Expected Rate of Return

ps =Price

Dividend=

16.42$

95.1$= .0463, or 4.63%

8-5A. (a) Expected return =Price

Dividend=

40$

40.3$= .085 = 8.5%

(b) Given your 8 percent required rate of return, the stock is worth $42.50 to you.

Value =Returnof RateRequired

Dividend=

08.

40.3$= $42.50

Since the expected rate of return (8.5%) is greater than your required rate of return (8%), or since the current market price ($40) is less than the value($42.50), the stock is undervalued and you should buy.

211



8-6A. Value (Vcs) =Rate)Required(1

1Year inDividend

++

Rate)Required(1

1Year inPrice

+

$50 =)15.1(

6$

++

)15.1(

P1

+

Rearranging and solving for P1:

P1 = $50 (1.15) - $6

P1 = $51.50

The stock would have to increase $1.50 ($51.50 - $50) or 3 percent($1.50/$50) to earn a 15% rate of return.

8-7A. (a))k (returnof

rateExpected

cs

=PriceMarket

1Year inDividend+

rategrowth

cs

=

50.22$

00.2$+ .10

cs = .1889, or 18.9%

(b) Vcs =10. 17.

00.2$

−= $28.57

Yes, purchase the stock. The expected return is greater than your requiredrate of return. Also, the stock is selling for only $22.50, while it is worth$28.57 to you.

8-8A. Value (Vcs) =Rate)GrowthRate(Required

Rate)Growth(1DividendYear Last

−+

Vcs =

Vcs = $24.50

8-9A. Growth rate = return on equity x retention rate

= (18%) × (40%) = 7.2%

8-10A. Expected Rate of Return ( csk ) =Price

Rate)Growth(1DividendYear Last ++ Growth Rate

csk = + 0.095

csk = 0.193, or 19.3%

212



8-11A. Value (Vcs) =Rate)Required(1

1Year inDividend

++

Rate)Required(1

1Year inPrice

+

Vcs =)11.1(

85.1$+

)11.1(

50.42$

Vcs = $39.958-12A. If the expected rate of return is represented by csk :

Current Price =)k (1

1Year inDividend

cs++

)k (1

1Year inPrice

cs+

csk =PriceCurrent

1Year inPrice1Year inDividend +- 1

csk =00.43$

00.48$ 84.2$ +

- 1

csk = 0.1823, or 18.23%

8-13A.

(a) psk =Price

Dividend=

00.33$

60.3$

psk = 0.1091, or 10.91%

(b) Value (V ps) =Returnof RateRequired

Dividend=

10.0

60.3$= $36

(c) The investor's required rate of return (10 percent) is less than the expectedrate of return for the investment (10.91 percent). Also, the value of the stock to the investor ($36) exceeds the existing market price ($33), so buy thestock.

8-14A.(a) Expected Rate of Return =PriceMarket

1Year inDividend+ Rate

Growth

=50.23$

)08.1(32.1$+ 0.08

= 0.1407, or 14.07%

(b) Investor's Value =RateGrowth-Returnof RateRequired

1Year inDividend

=08.0105.0

)08.1(32.1$

−

= $57.02

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(c) Yes, the expected rate of return (14.07%) is greater than your required rate of return (10.5 percent). Also, your value of the stock ($57.02) is greater thanthe current market price ($23.50).

8-15A (a) Dividend yield: Dividend ÷ stock price =49$

12.1$= 0.0229, or 2.29%

(b) Using the nominal average returns of 12.2% for large-company stocks and the3.8% nominal average return for U.S. Treasury Bills as shown in Table 6-1,the computation would be as follows:

returnof rate

Expected=

rate

freerisk −+ beta ×

return

market-

−

rate

freerisk

= 3.8% + 1.10 × (12.2% - 3.8%) = 13.04%

(c)returnof rate

Expected=

PriceMarket


Growth

13.04% = 49$

12.1$

+ g

.1304 = .0229 + g

g = .1075, or 10.75%

8-16A

Johnson & Johnson

2003 2002 2001 2000 1999

EPS (diluted) $2.40 $2.16 $1.84 $1.61 $1.39Dividend $0.925 $0.795 $0.70 $0.62 $0.55

Stock Price (6/24/04): $55.75

Growth rate: $2.40 = $1.39 (1 + i)4

i = .1463, or 14.63%

k cs = gP

D

0

1 +

k cs =.1463

$55.75

(1.1463)$0.925+

k cs = .0190, or 1.90%

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8-17A.

(a) psk =Price

Dividend=

25$

50.4$

psk = 0.18, or 18%


Dividend =14.050.4$ = $32.14

(c) The investor's required rate of return (14 percent) is less than the expectedrate of return for the investment (18 percent). Also, the value of the stock tothe investor ($32.14) exceeds the existing market price ($25), so buy thestock.

8-18A.

(a) Investor's Value =RateGrowth-Returnof RateRequired

1Year inDividend

=05.015.0

)05.1(30.2$

−

= $24.15

(b) Expected Rate of Return =PriceMarket


Growth

=33$

)05.1(30.2$+ 0.05

= 0.1232, or 12.32%

(c) No, the expected rate of return (12.32%) is less than your required rate of return (15 percent). Also, your value of the stock ($24.15) is less than thecurrent market price ($33).

8-19A. (a) Growth rate = return on equity x retention rate

= (17%) × (30%) = 5.1%

(b) (i) If retention rate is 40%:

Growth rate = return on equity x retention rate

= (17%) × (40%) = 6.8%

(ii) If retention rate is 25%:


= (17%) × (25%) = 4.25%

215



8-20A. (a))k (returnof

rateExpected

cs

=PriceMarket

1Year inDividend+

rategrowth

cs =50.29$

)04.01(75.1$ ++ 0.04

cs = .1017, or 10.17%

(b) Vcs =04.0 14.

)04.01(75.1$

−+

= $18.20

No, do not purchase the stock. The expected return is less than your requiredrate of return. Also, the stock is selling for $29.50, while it is only worth$18.20 to you.

SOLUTION TO INTEGRATIVE PROBLEM

1. Value (V b) based upon your required rate of return:

Bond:

V b =t

12

1t .12)(1

$140

+∑=

+12)12.1(

000,1$

+

12

12

140

1000

CPT → ANSWER -1123.89

Preferred Stock:

V ps =t

1t .14)(1

$12

+∑∞

=However, since the dividend is a constant amount each year with no maturitydate (infinity), the equation can be reduced to

V

ps

=

Returnof RateRequired

Dividend

=14.

12$

= $85.71

Common Stock:

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Step 1: Determine Growth Rate

$4.00 = ∑= +

++

10

1t10

bt

b )k 1(

00.8$

)k 1(

00.0$

10

40

8CPT → ANSWER 7.177%

Growth Rate (g) = 7.177%

Step 2: Solve for Value

Vcs =t

t

1t .20)(1.07177)$3(1 ++∑∞

=

If the growth rate (g) is assumed constant, the equation may be reduced to

Vcs =RateGrowthReturnof RateRequired

EndYear atDividend

−

=gk

D

cs

1

−

=07177.20.

)07177.1(3$

−+

= $25.08

2. Your Value Selling PriceBond $1,123.89 $1,200.00Preferred Stock 85.71 80.00Common Stock 25.08 25.00

The bond should not be purchased because its market value is selling aboveits value to you. You can choose between the preferred stock and the

common stock, because both have market values less than their values to you.

217



3. Bond:

Value (Vb) = ∑= +

++

12

1t12t .14) (1

$1,000

.14) (1

$140

1214

140

1000

CPT → ANSWER -1000.00

You would not buy the bond; it is not worth $1,200.00.

Preferred Stock:

V ps =16.

12$

= $75.00

Do not buy. Your value is less than what you would have to pay for thestock($80).

Common Stock:

Vcs =07177.18.

)07177.1(3$

−+

= $29.71

Buy. Your value is greater than what you would have to pay for the

stock($25).

4. Assuming a growth rate of 12 percent:

00.42$ 12.20.

)12.1(3 Vcs =

−+

=

Buy the stock. Because of the expected increase in future dividends, thestock is now worth more to you ($42) than you would have to pay for it($25) –assuming that the selling price did not increase also.

218



Solutions to Problem Set B

8-1B. Value (V ps) =10.

7$

= $70.00

8-2B. Growth rate = return on equity x retention rate

= (24%) × (70%) = 16.8%

8-3B. Value (V ps) =12.

100$ 16. ×

=12.

16$

= $133.33

8-4B. Expected Rate of Return

ps =Price

Dividend =16.55$

35.2$ = .0426, or 4.26%

8-5B. (a) Expected return =Price

Dividend=

50.38$

25.3$= .0844 , or 8.44%

(b) Given your 8 percent required rate of return, the stock is worth $40.63 to you.

Value =Returnof RateRequired

Dividend=

08.

25.3$= $40.63

Since the expected rate of return (8.44%) is greater than your required rate of

return (8%), or since the current market price ($38.50) is less than the value($40.63), the stock is undervalued and you should buy.

8-6B. Value (Vcs) =Rate)Required (1

1Year inDividend

++

Rate)Required (1

1Year inPrice

+

$52.75 =)16.1(

50.6$

++

)16. 1(

P1

+

Rearranging and solving for P1:

P1 = $52.75 (1.16) - $6.50P1 = $54.69

The stock would have to increase $1.94 ($54.69 - $52.75), or 3.68 percent,($1.94/$52.75) to earn a 16% rate of return.

219



8-7B. (a))k (returnof

rateExpected

cs

=PriceMarket


Growth

csk =00.23$

50.2$+ .105

csk = 0.2137, or 21.37%

(b) Vcs =105.17.

50.2$

−= $38.46

The expected rate of return exceeds your required rate of return, which meansthat the value of the security to you is greater than the current market price.Thus, you should buy the stock.

8-8B. Value (Vcs) =Rate)GrowthRate(Required

Rate)Growth(1DividendYear Last

−+

V

cs

=

06.20.

)06.1(75.3$

−

+

Vcs = $28.39

8-9B. Growth rate = return on equity x retention rate

= (24%) × (60%) = 14.4%

8-10B. Expected Rate of Return ( csk )=Price

Rate)Growth(1DividendYear Last ++ Growth

Rate

csk =

84.33$

)085.1(00.3$+ 0.085 = 0.1812, or 18.12%

8-11B. Value (Vcs) =Rate)Required (1

1Year inDividend

++

Rate)Required (1

1Year inPrice

+

Vcs =)12.1(

85.1$+

)12.1(

00.40$

Vcs = $37.37

8-12B. If the expected rate of return is represented by csk :

Current Price = )k (1

1Year inDividend

cs+ + )k (1

1Year inPrice

cs+

csk =PriceCurrent

1Year inPrice 1Year inDividend +- 1

csk =00.44$

00.47$ 00.2$ +- 1

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csk = 0.1136, or 11.36%

8-13B. (a) psk =Price

Dividend=

00.35$

00.4$= 11.43%

(b) Value (V ps ) =Returnof RateRequired

Dividend=

10.0

00.4$= $40

(c) The investor's required rate of return (10 percent) is less than the expectedrate of return for the investment (11.43 percent). Also, the value of the stock to the investor ($40) exceeds the existing market price ($35). The investor should buy the stock.

8-14B. (a) Expected Rate of Return =PirceMarket

1Year inDividend+

=00.25$

)08.1(00.1$+ 0.08

= 0.1232, or 12.32%

(b) Investor's Value =RateGrowthReturnof RateRequired

1Year inDividend

−

=08.011.0

)08.1(00.1$

−

= $36.00(c) Yes, the expected rate of return is greater than your required rate of return

(12.32 percent versus 11 percent). Also, your value of the stock ($36.00) ishigher than the current market price ($25.00).

8-15B (a) Dividend yield: Dividend ÷ stock price =54$20.1$ = 2.22%

(b) Using the nominal average returns of 12.2% for large-company stocks and the3.8% nominal average return for U.S. Treasury Bills as shown in Table 6-1,the computation would be as follows:

returnof rateExpected

=rate

freerisk −+ beta ×

return

market-

−

rate

freerisk

= 3.8% + 0.90 × (12.2% - 3.8%) = 11.36%

(c) returnof rateExpected

= PriceMarket


Growth

11.36% =54$

20.1$+ g

.1136 = .0222 + g

g = .0914, or 9.14%

221



8-16B

Note to Instructor: Before the 10th edition of Financial Management was completed, wehad not verified the earnings per share data for First Union Corporation. After the text wentto production, we realized that First Union was merged into Wachovia Bank in 2001. Wethen planned to use the Wachovia earnings per share data, only to discover the significantvolatility of the firm’s earnings over the past five years, which appears as follows:

2003 2002 2001 2000 1999

Earnings per share (diluted) $3.18 $2.60 $1.45 $0.07 $3.33

Since Wachovia’s earnings per share has actually decreased over the last five years, thegrowth rate in earnings per share is =1.146 percent (using 4 years to compute the growthrate). Clearly, the historical earnings per share do not provide a reasonable estimate of futureearnings per share, which regrettably makes the problem difficult to use without having ameaningful growth estimate. Thus, the problem can only be used to show that relying onhistorical data does not always provide reasonable results.

8-17B. (a) psk =Price

Dividend =50.19$25.2$

psk = 0.1154, or 11.54%


Dividend=

10.0

25.2$= $22.50

(c) The investor's required rate of return (10 percent) is less than the expectedrate of return for the investment (11.54 percent). Also, the value of the stock to the investor, ($22.50) exceeds the existing market price ($19.50), so buythe stock.

8-18B.

(a) Investor's Value =RateGrowth-Returnof RateRequired

1Year inDividend

=05.012.0

)05.1(95.1$

−

= $29.25

(b) Expected Rate of Return =PriceMarket


Growth

=26$

)05.1(95.1$+ 0.05

= 0.1288, or 12.88%

(c) Yes, the expected rate of return (12.88%) is greater than your required rateof return (12 percent). Also, your value of the stock ($29.25) is greater thanthe current market price ($26).

222



8-19B. (a) Growth rate = return on equity x retention rate

= (13%) × (20%) = 2.6%

(b) (i) If retention rate is 35%:


= (13%) × (35%) = 4.55%

(ii) If retention rate is 13%:


= (13%) × (13%) = 1.69%

8-20B. (a))k (returnof

rateExpected

cs

=PriceMarket

1Year inDividend+

rategrowth

cs =75.33$

)07.01(15.3$ ++ 0.07

cs = .1699, or 16.99%

(b) Vcs =07.0 11.

)07.01(15.3$

−+

= $84.26

Yes, purchase the stock. The expected return is significantly more than your required rate of return. Also, the stock is selling for $33.75, while it is worth$84.26 to you.

Solutions to Appendix 8A

8A-1. Using the NVDG model,

Vcs =cs

1

k

EPS+

gk

PV

cs

1

−

where k cs = the investor's required rate of return

EPS1 = the firm's earning per share in year 1

g = the growth rate, which is the firm's earnings retention ratetimes its return on equity.

PV1 =

cs

x1

k

ROEEPSr x- (r x EPS1)

r = the firm's earnings retention rate

ROE = the firm's return on equity investment

223



For our problem,

PV1 =

16.0

)20.0(x)5($x)65.0(- (0.65 x $5)

= $4.0625 - $3.25

= $0.8125

and Vcs =)20.0)(65.0(16.

8125.0$

16.

5$

−+

= $31.25 + $27.08

= $58.33

Using the more traditional dividend-growth model, we get:

Vcs = g k

D

cs

1

−

Since D1 = EPS1(1 - the retention rate), and

g = the retention rate x return on equity

Vcs =)20)(.65(.16.

)65.1)(5($

−−

=03.

75.1$= $58.33

8A-2. Given the EPS1 is expected to be $7 and the investor's required rate of return is 18

percent, the value of the stock, assuming no growth opportunities would be:

Vcs

=18.

7$

k

EPS

cs

1

== $38.89

where k cs = the investor's required rate of return

EPS1 = the firm's earning per share in year 1

To compute the present value of the growth opportunities, NVDG, for each scenario,

we use the following equation:

NVDG =gk

PV

cs

1

−

where PV1 =

cs

x1

k

ROEEPSr x- (r x EPS1)

g = the growth rate, which is the firm's earnings retention ratetimes its return on equity.

r = the firm's earnings retention rate

ROE = the firm's return on equity investment

224



Given the different possible retention rates and ROEs, we may solve for therespective PV1s. The results are as follows:

Possible Different Retention Rates

ROEs 0% 30% 60%

16% 0.00 -0.23 -0.47

18% 0.00 0.00 0.00

24% 0.00 0.70 1.40

We next calculate the NVDG for each scenario by dividing the above PV1 values by

k cs - g, which gives the following results:


ROEs 0% 30% 60%

16% 0.00 -1.77 -5.56

18% 0.00 0.00 0.00

24% 0.00 6.48 38.89

Adding the $38.89 price, assuming no growth, to the above NVDGs, we get:


ROEs 0% 30% 60%

16% 38.89 37.12 33.33

18% 38.89 38.89 38.89

24% 38.89 45.37 77.78

Thus, our results show that value is created only when management reinvests atabove the investor's required rate of return. That is, growth may actually decrease the

firm's value if the profitability of the new investments are not adequate enough to

satisfy the investor's required returns.

chapter 08 im 10th ed

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